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February  2019, 13(1): 197-210. doi: 10.3934/ipi.2019011

Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators

1. 

Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

2. 

Department of Mathematics and Statistics, University of Jyväskylä, Jyväskylä, Finland

3. 

Department of Mathematics, Hong Kong Baptist University, Kowloon, Hong Kong SAR, China

Received  May 2018 Revised  October 2018 Published  December 2018

Let $A∈{\rm{Sym}}(n× n)$ be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator $\mathscr{L}_A^s+q$, where $\mathscr{L}_A^s: = (-\nabla·(A(x)\nabla))^s$, $s∈ (0, 1)$ and $q∈ L^∞$. We are concerned with the simultaneous recovery of $q$ and possibly embedded soft or hard obstacles inside $q$ by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain $Ω$ associated with $\mathscr{L}_A^s+q$. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential $q$. If multiple measurements are allowed, then the surrounding potential $q$ can also be uniquely recovered. These are surprising findings since in the local case, namely $s = 1$, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are long-standing problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.

Citation: Xinlin Cao, Yi-Hsuan Lin, Hongyu Liu. Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators. Inverse Problems & Imaging, 2019, 13 (1) : 197-210. doi: 10.3934/ipi.2019011
References:
[1]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single farfield measurement, Proc. Amer. Math. Soc., 133 (2005), 1685–1691. Corrigendum: Preprtint arXiv math.AP/0601406, 2006. doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[2]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

[3]

J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 19 (2003), 1361-1384.  doi: 10.1088/0266-5611/19/6/008.  Google Scholar

[4]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[5]

T. GhoshY.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communications in Partial Differential Equations, 42 (2017), 1923-1961.  doi: 10.1080/03605302.2017.1390681.  Google Scholar

[6]

T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv: 1609.09248. Google Scholar

[7]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641. Google Scholar

[8]

N. HondaG. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Math. Ann., 355 (2013), 401-427.  doi: 10.1007/s00208-012-0786-0.  Google Scholar

[9]

O. ImanuvilovG. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Am. Math. Soc., 23 (2010), 655-691.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[10]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer-Verlag, New York, 2006.  Google Scholar

[11]

C. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math.(2), 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[12]

A. Kirsch X. Liu, Direct and inverse acoustic scattering by a mixed-type scatterer, Inverse Problems, 29 (2013), 065005, 19pp. doi: 10.1088/0266-5611/29/6/065005.  Google Scholar

[13]

A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 619-651.  doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.  Google Scholar

[14]

R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, arXiv: 1710.07404. Google Scholar

[15]

H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data, Inverse Problems, 33 (2017), 065001, 20pp. doi: 10.1088/1361-6420/aa6770.  Google Scholar

[16]

H. LiuM. PetriniL. Rondi and J. Xiao, Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differential Equations, 262 (2017), 1631-1670.  doi: 10.1016/j.jde.2016.10.021.  Google Scholar

[17]

H. LiuH. Zhao and C. Zou, Determining scattering support of anisotropic acoustic mediums and obstacles, Commun. Math. Sci., 13 (2015), 987-1000.  doi: 10.4310/CMS.2015.v13.n4.a7.  Google Scholar

[18]

X. Liu and B. Zhang, Direct and inverse obstacle scattering problems in a piecewise homogeneous medium, SIAM J. Appl. Math., 70 (2010), 3105-3120.  doi: 10.1137/090777578.  Google Scholar

[19]

H. Liu and J. Zou, Uniqueness in an inverse acoustic scatterer scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[20]

H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far-field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016.  Google Scholar

[21]

H. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, J. Phys.: Conf. Ser., 124 012006. Google Scholar

[22]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.  Google Scholar

[23]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[24]

S. O'Dell, Inverse scattering for the Laplace-Beltrami operator with complex electromagnetic potentials and embedded obstacles, Inverse Problems, 22 (2006), 1579-1603.  doi: 10.1088/0266-5611/22/5/005.  Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Volume 44, Springer Science & Business Media, 2012. Google Scholar

[26]

L. Rondi, Unique determination of non-smooth sound-soft scatterers by finitely many far-field measurements, Indiana Univ. Math. J., 52 (2003), 1631-1662.  doi: 10.1512/iumj.2003.52.2394.  Google Scholar

[27]

L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J., 57 (2008), 1377-1408.  doi: 10.1512/iumj.2008.57.3217.  Google Scholar

[28]

W. Rudin, Functional Analysis, New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[29]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21 pp, arXiv: 1711.04799. doi: 10.1088/1361-6420/aaac5a.  Google Scholar

[30]

A. Rüland and M. Salo, The fractional Calderón problem: low regularity and stability, arXiv: 1708.06294. Google Scholar

[31]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Communications in Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

show all references

References:
[1]

G. Alessandrini and L. Rondi, Determining a sound-soft polyhedral scatterer by a single farfield measurement, Proc. Amer. Math. Soc., 133 (2005), 1685–1691. Corrigendum: Preprtint arXiv math.AP/0601406, 2006. doi: 10.1090/S0002-9939-05-07810-X.  Google Scholar

[2]

L. A. Caffarelli and P. R. Stinga, Fractional elliptic equations, Caccioppoli estimates and regularity, Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 33 (2016), 767-807.  doi: 10.1016/j.anihpc.2015.01.004.  Google Scholar

[3]

J. Cheng and M. Yamamoto, Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves, Inverse Problems, 19 (2003), 1361-1384.  doi: 10.1088/0266-5611/19/6/008.  Google Scholar

[4]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-03537-5.  Google Scholar

[5]

T. GhoshY.-H. Lin and J. Xiao, The Calderón problem for variable coefficients nonlocal elliptic operators, Communications in Partial Differential Equations, 42 (2017), 1923-1961.  doi: 10.1080/03605302.2017.1390681.  Google Scholar

[6]

T. Ghosh, M. Salo and G. Uhlmann, The Calderón problem for the fractional Schrödinger equation, arXiv: 1609.09248. Google Scholar

[7]

B. Harrach and Y.-H. Lin, Monotonicity-based inversion of the fractional Schrödinger equation, arXiv: 1711.05641. Google Scholar

[8]

N. HondaG. Nakamura and M. Sini, Analytic extension and reconstruction of obstacles from few measurements for elliptic second order operators, Math. Ann., 355 (2013), 401-427.  doi: 10.1007/s00208-012-0786-0.  Google Scholar

[9]

O. ImanuvilovG. Uhlmann and M. Yamamoto, The Calderón problem with partial data in two dimensions, J. Am. Math. Soc., 23 (2010), 655-691.  doi: 10.1090/S0894-0347-10-00656-9.  Google Scholar

[10]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer-Verlag, New York, 2006.  Google Scholar

[11]

C. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math.(2), 165 (2007), 567-591.  doi: 10.4007/annals.2007.165.567.  Google Scholar

[12]

A. Kirsch X. Liu, Direct and inverse acoustic scattering by a mixed-type scatterer, Inverse Problems, 29 (2013), 065005, 19pp. doi: 10.1088/0266-5611/29/6/065005.  Google Scholar

[13]

A. Kirsch and L. Päivärinta, On recovering obstacles inside inhomogeneities, Math. Meth. Appl. Sci., 21 (1998), 619-651.  doi: 10.1002/(SICI)1099-1476(19980510)21:7<619::AID-MMA940>3.0.CO;2-P.  Google Scholar

[14]

R.-Y. Lai and Y.-H. Lin, Global uniqueness for the semilinear fractional Schrödinger equation, arXiv: 1710.07404. Google Scholar

[15]

H. Liu and X. Liu, Recovery of an embedded obstacle and its surrounding medium from formally determined scattering data, Inverse Problems, 33 (2017), 065001, 20pp. doi: 10.1088/1361-6420/aa6770.  Google Scholar

[16]

H. LiuM. PetriniL. Rondi and J. Xiao, Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements, J. Differential Equations, 262 (2017), 1631-1670.  doi: 10.1016/j.jde.2016.10.021.  Google Scholar

[17]

H. LiuH. Zhao and C. Zou, Determining scattering support of anisotropic acoustic mediums and obstacles, Commun. Math. Sci., 13 (2015), 987-1000.  doi: 10.4310/CMS.2015.v13.n4.a7.  Google Scholar

[18]

X. Liu and B. Zhang, Direct and inverse obstacle scattering problems in a piecewise homogeneous medium, SIAM J. Appl. Math., 70 (2010), 3105-3120.  doi: 10.1137/090777578.  Google Scholar

[19]

H. Liu and J. Zou, Uniqueness in an inverse acoustic scatterer scattering problem for both sound-hard and sound-soft polyhedral scatterers, Inverse Problems, 22 (2006), 515-524.  doi: 10.1088/0266-5611/22/2/008.  Google Scholar

[20]

H. Liu and J. Zou, On unique determination of partially coated polyhedral scatterers with far-field measurements, Inverse Problems, 23 (2007), 297-308.  doi: 10.1088/0266-5611/23/1/016.  Google Scholar

[21]

H. Liu and J. Zou, On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems, J. Phys.: Conf. Ser., 124 012006. Google Scholar

[22]

W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.  Google Scholar

[23]

E. D. NezzaG. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des Sciences Mathématiques, 136 (2012), 521-573.  doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar

[24]

S. O'Dell, Inverse scattering for the Laplace-Beltrami operator with complex electromagnetic potentials and embedded obstacles, Inverse Problems, 22 (2006), 1579-1603.  doi: 10.1088/0266-5611/22/5/005.  Google Scholar

[25]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Volume 44, Springer Science & Business Media, 2012. Google Scholar

[26]

L. Rondi, Unique determination of non-smooth sound-soft scatterers by finitely many far-field measurements, Indiana Univ. Math. J., 52 (2003), 1631-1662.  doi: 10.1512/iumj.2003.52.2394.  Google Scholar

[27]

L. Rondi, Stable determination of sound-soft polyhedral scatterers by a single measurement, Indiana Univ. Math. J., 57 (2008), 1377-1408.  doi: 10.1512/iumj.2008.57.3217.  Google Scholar

[28]

W. Rudin, Functional Analysis, New York-Düsseldorf-Johannesburg, 1973.  Google Scholar

[29]

A. Rüland and M. Salo, Exponential instability in the fractional Calderón problem, Inverse Problems, 34 (2018), 045003, 21 pp, arXiv: 1711.04799. doi: 10.1088/1361-6420/aaac5a.  Google Scholar

[30]

A. Rüland and M. Salo, The fractional Calderón problem: low regularity and stability, arXiv: 1708.06294. Google Scholar

[31]

P. R. Stinga and J. L. Torrea, Extension problem and Harnack's inequality for some fractional operators, Communications in Partial Differential Equations, 35 (2010), 2092-2122.  doi: 10.1080/03605301003735680.  Google Scholar

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